Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 306 / 1 / c / ii

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- Solution ii

Solution

For the circular embedding,

γ_{ττ} = - 1 + \dot{R}^{2}, γ_{τ σ} = 0, γ_{σσ} = R^{2} .

On

R = R_{0} cos (τ / R_{0})

this becomes

(γ_{α β}) = cos^{2} (\frac{τ}{R _{0}}) (- 1 0 0 R_{0}^{2}) .

Thus the worldsheet metric is conformally flat between collapse events and becomes degenerate at the instant when the loop shrinks to zero size.

Solved by gpt-5.6-sol high.

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